Edit: I also find it funny when people get passionate about their "free" country's national anthems when they're one bad underpaid, undereducated border patrol agent away from not having any rights.
:)
According to the "Port Courtesy Handbook" it sounds like many foreign dignitaries go through a modified and expedited border check at the US border that the rest of us do not use. "A Port Courtesy or “Courtesy of the Port” provides Foreign Government Officials and their traveling parties expedited processing and clearance upon arrival into the United States. Requests for Port Courtesies are managed by the Office of the Chief of Protocol in coordination with U.S. Customs and Border Protection (CBP)." https://www.state.gov/documents/organization/170352.pdf
And anyway, my understanding is that if you're rich enough to charter a private jet, or to own one, you skip a lot of the border formalities.
Perhaps if you carry a diplomatic passport, that's a different story altogether.